# Characteristic subgroup of solvable group

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property imposed on theambient group: solvable group

View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

## Contents

## Definition

A subgroup in a group is termed a **characteristic subgroup of solvable group** if is a solvable group and is a characteristic subgroup.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

characteristic subgroup of nilpotent group | the subgroup is characteristic and the whole group is a nilpotent group | follows from nilpotent implies solvable | solvable not implies nilpotent -- use the group as a subgroup of itself | |FULL LIST, MORE INFO |

characteristic subgroup of abelian group | the subgroup is characteristic and the whole group is an abelian group | (via nilpotent) | (via nilpotent) -- use the group as a subgroup of itself | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

normal subgroup of solvable group | characteristic implies normal | normal not implies characteristic in any nontrivial subquasivariety of the quasivariety of groups plus solvability is quasivarietal | |FULL LIST, MORE INFO | |

solvable characteristic subgroup | the subgroup is characteristic and solvable as a group in its own right. | solvability is subgroup-closed | the trivial subgroup in any non-solvable group. | |FULL LIST, MORE INFO |

solvable normal subgroup | (via solvable characteristic, also via normal subgroup of solvable group) | (via solvable characteristic, also via normal subgroup of solvable group) | |FULL LIST, MORE INFO |